PERFORMANCE OF APN GUIDANCE LAW IN A TWO DIMENSIONAL
AERIAL ENGAGEMENT SCENARIO
K.N.Swamy, Scientist, Advanced Technology Programme and
I.G.Sarma, Professor of Computer Science and Automation
Indian Institute of Science, Bangalore, India.
in P N , b u t f o r t h e e x t r a a d d i t i v e t e r m 1/2 "aT
w h e r e , N' is t h e e f f e c t i v e n a v i g a t i o n r a t i o a n d
aT is t h e t a r g e t a c c e l e r a t i o n .
The t a r g e t and
t h e missile a r e b o t h a s s u m e d to h a v e a c e l e r a t i o n s
normal t o t h e chosen reference direction.
In
t h e a c t u a l i m p l e m e n t a t i o n of APN, t h e r e is t h e
a d d i t i o n a l c o m p l e x i t y of having to e s t i m a t e t h e
t a r g e t m a n e u v e r o n b o a r d t h e missile. T h e t a r g e t
m o d e l s t h a t a r e g e n e r a l l y u s e d in l i t e r a t u r e a r e
t h o s e o b t a i n e d by passing w h i t e noise t h r o u g h
a shaping f i l t e r [3,41.
A step t a r g e t maneuver
w i t h t h e t i m e of i n i t i a t i o n uniformly d i s t r i b u t e d
o v e r t h e e n g a g e m e n t t i m e is shown t o c o r r e s p o n d
t o w h i t e noise t h r o u g h a n i n t e g r a t o r [5].
Abstract
In t h e a v a i l a b l e l i t e r a t u r e , t h e m a i n a d v a n t a g e
c l a i m e d f o r APN is t h a t i t d e m a n d s a p r o g r e s s i v e l y
d e c r e a s i n g missile a c c e l e r a t i o n in t h e c o n s t a n t
t a r g e t m a n e u v e r s i t u a t i o n , as o p p o s e d to a n inc r e a s i n g missile a c c e l e r a t i o n f o r t h e m o r e c o n v e n t i o n a l PN guidance.
This i n f e r e n c e is mainly
b a s e d o n s t u d i e s w i t h o n e d i m e n s i o n a l models.
The present study, based on a two dimensional
m o d e l , n o t only i n c o r p o r a t e s a m o r e r e a l i s t i c t a r g e t
m a n e u v e r , b u t a l s o c o n s i d e r s t h e effect of t a r g e t
g l i n t c h a r a c t e r i s e d by c o r r e l a t e d noise. T h e missile
acceleration
a n d t h e m i s s - d i s t a n c e i n d u c e d by
t a r g e t maneuver and glint a r e t h e two features
of p e r f o r m a n c e i n v e s t i g a t e d in t h i s s t u d y .
In t h e
deterministic study, with target maneuver initiation
o c c u r i n g a t v a r i o u s p o i n t s in t i m e , t h e a c c e l e r a t i o n
profiles resemble t h e ones obtained from a linear
q u a d r a t i c t r a c k i n g f o r m u l a t i o n of t h e g u i d a n c e
problem. T h e r e s u l t s i n d i c a t e t h a t t h e r e is a consid e r a b l e i m p r o v e m e n t in t h e miss d i s t a n c e p e r f o r mance over PN, for maneuvers which a r e initiated
w h e n t h e r e l a t i v e s e p e r a t i o n is small.
The present study assumes a deterministic
t a r g e t m a n e u v e r m o d e l , r e p r e s e n t e d by i n i t i a l
l e v e l f l i g h t , f o l l o w e d by t h e e x e c u t i o n of a c o n s t a n t
a c c e l e r a t i o n m a n e u v e r in t h e v e r t i c a l plane.
F r o m a m o n g t h e v a r i o u s s o u r c e s of noise w h i c h
affect t h e p e r f o r m a n c e , t h e effect of t a r g e t
g l i n t c h a r a c t e r i s e d by c o r r e l a t e d noise is investig a t e d , s i n c e i t is t h e m o s t s i g n i f i c a n t . T h e missile
a c c e l e r a t i o n a n d t h e miss-distance induced by
t a r g e t maneuver and glint a r e t h e t w o features
of p e r f o r m a n c e i n v e s t i g a t e d in t h i s study.
1. Introduction
The studies have been carried-out under
t h e following conditions: o n e in which t h e t a r g e t
maneuver level and its initiation t i m e a r e assumed
t o be known p r e c i s e l y a n d t h e o t h e r , w h e r e i n
t h e t a r g e t m a n e u v e r is e s t i m a t e d by a w e i g h t e d
least squares method with t h e attendant detection
delay and estimation error.
In a d d i t i o n t o P N ,
a version of L i n e a r Q u a d r a t i c (LQ) g u i d a n c e
c a l l e d MAMS [61 is a l s o u s e d f o r c o m p a r i n g t h e
performance. The comparative acceleration profiles
a r e d i s c u s s e d for P N , APN a n d L Q g u i d a n c e
s c h e m e s , a n d t h e miss d i s t a n c e s f o r t h e PN a n d
APN s c h e m e s .
C l a s s i c a l g u i d a n c e l a w s like t h e P r o p o r t i o n a l
Navigation (PN) have been employed quite successfully in t h e p a s t f o r homing guidance.
In p a r t i c u l a r , using low p a s s f i l t e r i n g t o a t t e n u a t e t h e
noise i n h e r e n t in t h e g u i d a n c e signal a n d applying
t h e P N l a w t o s t e e r t h e missile t o w a r d s t h e t a r g e t
w o r k s well in benign e n v i r o n m e n t s .
However,
in r e c e n t y e a r s t h e p e r f o r m a n c e of t h r e a t a i r c r a f t
h a v e i m p r o v e d s u b s t a n t i a l l y in t e r m s of s p e e d
a n d agility. T h e p e r f o r m a n c e i m p r o v e m e n t r e q u i r e d
of PN to m e e t t h e s e a d d i t i o n a l d e m a n d s i n c r e a s e s
t h e a c c e l e r a t i o n r e q u i r e m e n t of t h e a i r f r a m e .
While t h e r e is a g r e a t d e a l of published l i t e r a t u r e
o n P N , t h a t a v a i l a b l e o n A u g m e n t e d PN ( A P N )
is s c a n t y .
T h e e x h a u s t i v e s u r v e y p a p e r by P a s t r i c k
e t & [ I ] w h i l e discussing PN a t l e n g t h , h a r d l y
p r o v i d e s any i n f o r m a t i o n a b o u t APN.
2. Models for target maneuver and glint
T h e c o m p a r a t i v e performance study for
t h e t h r e e g u i d a n c e s c h e m e s r e q u i r e s developm e n t a n d u s e of s u i t a b l e m o d e l s f o r t a r g e t m a n e u v e r a n d glint. T h e t a r g e t m a n e u v e r m o d e l proposed
h e r e is d e t e r m i n i s t i c involving a c o n s t a n t s t e p
a c c e l e r a t i o n in t h e v e r t i c a l p l a n e till it l e v e l s
off f o r r e t r e a t . On t h e o t h e r hand, b e i n g a noise
t e r m , g l i n t is m o d e l l e d s t o c h a s t i c a l l y .
T h e m a i n a d v a n t a g e c l a i m e d f o r APN by N e s l i n e
a n d Z a r c h a n [2] is t h a t it d e m a n d s a progressively
d e c r e a s i n g missile a c c e l e r a t i o n in t h e c o n s t a n t
t a r g e t m a n e u v e r s i t u a t i o n , as o p p o s e d to a n inc r e a s i n g missile a c c e l e r a t i o n f o r t h e m o r e c o n v e n t i o n a l PN guidance.
This is a c h i e v e d by e x p l i c i t l y
i n c o r p o r a t i n g t h e t a r g e t a c e l e r a t i o n t e r m in t h e
APN g u i d a n c e law. In o t h e r words, t h e c o m m a n d e d
a c c e l e r a t i o n in APN h a s t h e s a m e f o r m a s t h a t
Target maneuver models
T h e p r o b l e m of how t o m o d e l t a r g e t a c c e l e r a -
209
CH2759-9/89/0000-0209 $1 .OO
0
1989 IEEE
given by
t i o n t o a c h i e v e i m p r o v e d missile p e r f o r m a n c e ,
h a s b e e n e x t e n s i v e l y discussed in l i t e r a t u r e .
This is a c r i t i c a l problem a r e a f o r APN a n d
L Q guidance implementations, since t h e estimat i o n of
missile-to-target
position,
velocity
a n d a c c e l e r a t i o n d e p e n d o n this.
a2 =
3. PN Guidance
Traditionally,
homing i n t e r c e p t p r o b l e m s
h a v e b e e n solved t h r o u g h t h e u s e of PN. This
f o r m of g u i d a n c e l a w c o m m a n d s a missile
a c c e l e r a t i o n which is p r o p o r t i o n a l t o t h e line-of-sight (LOS) r a t e a n d t h e proportionality f a c t o r
is c a l l e d t h e Navigation r a t i o . T h e e n g a g e m e n t
g e o m e t r y is shown in Fin.2.
The commanded
a c c e l e r a t i o n is given by
a = N'V
In t h e t a r g e t model proposed and used
in t h i s study, i t is a s s u m e d t h a t t h e t a r g e t
f l i e s level till t i m e t m a n ,a n d t h e n e x e c u t e s
a c o n s t a n t m a n e u v e r aT till i t levels-off f o r
a r e t r e a t as shown in Fig. 1.
This model
is j u s t i f i e d f r o m p r a c t i c a l considerations,
s i n c e t h e t a r g e t generally pulls a m a x i m u m
m a n e u v e r a n d r e t r e a t s a f t e r delivery of i t s
weapon load.
The m a n e u v e r is d e t e c t e d a n d
e s t i m a t e d on board t h e missile a f t e r its occurrence a t t
The m a g n i t u d e of t a r g e t veloc i t y VT is %%u";ned t o b e a c o n s t a n t .
ACCELERATION
iZ
c
V
zl.
TRAJECTORY
TARGET
FIG.
1
= Closing v e l o c i t y and
N' = E f f e c t i v e n a v i g a t i o n r a t i o
/
b
TIME
I
X
where
I
I.
c
I
/
M
(bl
1
LEVEL AND STEP TARGET MANEbVER
c
01
FIG. 2
Glint Noise
+g E2T9 a9 '
consnoise
these
3met.
GEOMETRY
Choice of e f f e c t i v e navigation ratio
T h e p e r f o r m a n c e of t h e missile is f i r s t
c o n s i d e r e d a g a i n s t m a n e u v e r i n g t a r g e t s for
different maneuver initiation times, for values
of n a v i g a t i o n r a t i o N' in t h e r a n g e 3 t o 5.
R e s u l t s f o r t h e d e t e r m i n i s t i c case i n d i c a t e
t h a t t h e miss d i s t a n c e s d e c r e a s e with i n c r e a s i n g
N'.
T h e missile a c c e l e r a t i o n s a r e l a r g e r initially f o r N ' = 5 c o m p a r e d toN':3 o r 4. A c o m p a r i s o n of r e s u l t s f o r various N' in t h e r a n g e
4 t o 8 f o r t h e s t o c h a s t i c case ( w i t h g l i n t noise
a n d f i l t e r ) r e v e a l s t h a t t h e c o m m a n d e d acce l e r a t i o n b e c o m e s very l a r g e a n d noisy f o r
N'=8, essentially
d u e t o t h e amplification
of g l i n t noise.
Based on t h e a b o v e studies,
a c o m p r o m i s e v a l u e of 5 is c h o s e n f o r N'.
Following [71, t o g e r a t e w h i t e noise
of s p e c t r a l d e n s i t y ( 2 T g Z g ) , a w h i t e s e q u e n c e
of v a r i a n c e 6 2 is g e n e r a t e d w i t h a spacing
of A
Assuming t h a t t h e c o r r e l a t i o n of t h e
w h i t e s e q u e n c e drops t o z e r o o v e r A , a triangul a r c o r r e l a t i o n is obtained. S i n c e f o r s m a l l
A , s p e c t r a l density is a p p r o x i m a t e l y t h e a r e a
under t h e correlation curve, we have for t h e
white sequence generated
.
Spectral density$
ENGAGEMENT
T h e principal o u t p u t s of t h e missile s e e k e r
a r e t h e dish r a t e which is a m e a s u r e of t h e
line-of-sight r a t e , a n d t h e Doppler f r e q u e n c y
which is proportional t o t h e closing velocity.
T h e dish r a t e which is c o n t a m i n a t e d by glint
noise is passed through a noise smoothing
f i l t e r [2] t o o b t a i n t h e c o m m a n d e d a c c e l e r a t i o n
as shown in Fig.3.
G l i n t is c o r r e l a t e d noise which is r e p r e s e n t e d by passing w h i t e noise through a shaping
filter.
The e q u i v a l e n t w h i t e noise s p e c t r a l
d e n s i t y is g i v e n by [31
Where T g a n d 6 g a r e t h e c o r r e l a t i o n t i m e
t a n t and s t a n d a r d d e v i a t i o n of g l i n t
respectively.
R e p r e s e n t a t i v e values f o r
p a r a m e t e r s [3] a r e Tg = 0.1 sec. a n d 6 g =
X
=A&
Equating t h i s t o t h e e q u i v a l e n t g l i n t w h i t e
noise s p e c t r a l d e n s i t y , w e see t h a t t h e w h i t e
s e q u e n c e g e n e r a t e d will h a v e t h e v a r i a n c e
210
t o t h e r e f e r e n c e direction.
In t h e A P N guid a n c e e q u a t i o n developed h e r e , t h e c o m m a n d e d
missile a c c e l e r a t i o n of PN is a u g m e n t e d by
a n e x t r a t e r m which h a s X a n d 2 c o m p o n e n t s
of
t a r g e t a c c e l e r a t i o n resolved n o r m a l t o
t h e missile.
,
I
E ( aT, s?-aT
2
1s
- Seekor
P -
tn FIG. 3
Y
3
The APN Dynamic Model
?id
T h e t a r g e t t o missile r e l a t i v e
c o m p o n e n t s R I a n d R 3 a r e given by
time constant
1/15
Noise (Ilter time conslurit
separation
R1 = V I
SIMPLIFIED SCHEMATIC OF THE SEEKER LOOP
R 3 = V3
w h e r e VI a n d V3, t h e t a r g e t t o missile r e l a t i v e
velocity c o m p o n e n t s a r e g i v e n by
V1
Noise filter time constant
Since g l i n t noise h a s b e e n m o d e l e d as
a w h i t e noise p r o c e s s t h r o u g h a f i r s t o r d e r
f i l t e r of f r e q u e n c y 10 r a d / s e c , i t essentially
c o n t a i n s low f r e q u e n c y c o m p o n e n t s . A low-pass
f i l t e r of f r e q u e n c y 5 r a d i s e c ( T n = 0.2 s e c )
h a s b e e n i n t r o d u c e d to f i l t e r o u t t h e dish
r a t e noise. This value h a s b e e n c h o s e n a f t e r
studying t h e p e r f o r m a n c e of f i l t e r s with diffe r e n t t i m e c o n s t a n t s o v e r a r a n g e of 0.1 t o
0.5 sec. T h e m e a n miss d i s t a n c e s a t various
maneuver initiation t i m e s have been investigated. T h e s m a l l e s t miss d i s t a n c e h a s b e e n o b t a i n e d f o r a f i l t e r t i m e c o n s t a n t of 0.2 sec. The
f i l t e r w i t h T n = 0.2 sec, p e r m i t s t h e signal
t o pass t h r o u g h w i t h o u t much a t t e n u a t i o n
a n d c u t s off t h e noise e f f e c t i v e l y . F r o m t h e s e
considerations, t h e t i m e c o n s t a n t of 0.2 sec
is chosen.
V3
=
-a M SinyM
+
aT1
+ aT3
= aMCosyM
a T I a n d a T 3 a r e t h e c o m p o n e n t s of t a r g e t
a c c e l e r a t i o n , a n d aM IS t h e missile a c c e l e r a t i o n .
F r o m the engagement
t a n X = R 1)-
3
geometry
of
Fig.2.
1
Hence,
X = ( R
~
-V R ~~ v ~ ) /1( R3~ + R ~ )
Assuming t h a t t h e dish r a t e D I S proportional
t o t h e .angular e r r o r including glint, we g e t
D
-U (E,, +e8 I ,
w h e r e & b is t h e boresight e r r o r a n d c g is t h e
a n g u l a r glint noise a n d /4
is a c o n s t a n t ( / A =
‘&I.H e n c e
D = -/J[(D - A ) +E&
4. APN Guidance
Where D and
a r e t h e dish a n g l e and t h e
LOS a n g l e respectively. T h e flight p a t h a n g l e
d,,is g i v e n by
YM =
APN Law
T h e PN g u i d a n c e law
-
ac = NrV,X
aMflM
a n d t h e missile a c c e l e r a t i o n by
a,
c a n a l s o b e e x p r e s s e d as [2]
aC
-
-(-aM+ac)/Ta
a s s u m i n g a f i r s t o r d e r a u t o p i l o t lag of t i m e
c o n s t a n t ra
T h e a n g u l a r glint 6 , (Fig.4) is
given by
( N ‘/tgo2 ) ( Y + ? t g o )
.
w h e r e y = missile t a r g e t s e p a r a t i o n normal
t o r e f e r e n c e direction
R = Vctgo
Vc = closine. s p e e d ( c o n s t a n t )
E,
=
(-E,+
wn)
where
* Y/R,
E, * R
L
If t h e t a r g e t a c c e l a r a t i o n aT is considered,
the commanded accelaration becomes
+
T h e outpu; ?f t h e noise f i l t e r is given by
I
( D -K’/Twhich is a n e s t i m a t e of t h e LOS r a t e .
T h e expression f o r c o m m a n d e d a c c e l e r a t i o n
T h e g u i d a n c e law given a b o v e is t e r m e d as
A u g m e n t e d PN. This d i f f e r s f r o m t h e PN in
t h e inclusion of a n e x t r a t e r m d e p e n d e n t on
t a r g e t accelation. H e r e t h e relative separation,
r e l a t i v e velocity a n d t a r g e t a c c e l e r a t i o n a r e
c o n s i d e r e d only in o n e d i r e c t i o n ; n a m e l y n o r m a l
IS
a, =-N‘IvcI$
211
+ N?,
(a,Sin-),,-
3”)
~173c-s
In a c o m p a n i o n p a p e r by t h e a u t h o r s [6],
t h e r e f e r e n c e r e l a t i v e t r a j e c t o r y is d e r i v e d
f r o m a pulsed missile a c c e l e r a t i o n ,
which
h a s b e e n shown in l i t e r a t u r e t o b e t i m e a n d
energy optimal.
T h e w i d t h a n d l o c a t i o n of
t h e p u l s e d e p e n d s upon t h e e n g a g e m e n t conditions, heading e r r o r a n d t a r g e t m a n e u v e r .
T h e h e i g h t of t h e p u l s e c o u l d b e v a r i e d a c c o r d ing t o t h e l a t a x c a p a b i l i t y of t h e a i r f r a m e .
Heading e r r o r r e q u i r e s a n initial pulse, while
a constant g target maneuver demands another
r e f e r e n c e pulse, c o i n c i d i n g with t h e m a n e u v e r initiation.
T h e a b o v e set of e q u a t i o n s p r o v i d e t h e
t r a j e c t o r y f o r t h e stat% v a r i a b l e s RI,R3,V I , V 3 ,
E g a n d A , w h e r e a T , a n d aT 3
A,d,,,,D,aM,
a r e t h e c o m p o n e n t s of t a r g e t a c c e l e r a t i o n
obtained from a n assumed target trajectory
and
E g is t h e g l i n t noise. T h e c o m m a n d e d
a c c e l e r a t i o n ac is o b t a i n e d as a f u n c t i o n of
s t a t e variables.
-77
T
T h e v e h i c l e a c c e l e r a t i o n follows t h e r e f e r e n c e p u l s e closely in t h e o p t i m a l g u i d a n c e
scheme and t h e achieved acceleration remains
s m a l l f o r a l a r g e p o r t i o n of t h e f!ight t i m e .
6. Study Conditions
'b*g-
e
FIG. L
Unlike P N , b o t h APN a n d L Q g u i d a n c e
need a t a r g e t maneuver detection and estimation s c h e m e on b o a r d t h e missle, s i n c e t a r g e t
m a n e u v e r t e r m s a p p e a r e x p l i c i t l y in t h e guid a n c e a l g o r i t h m . A s t a t e m o d e l of planar
engagement with a mesurement model employing t h e homing h e a d a n d a u t o p i l o t m e a s u r e m e n t s
IS used
for maneuver estimation. A Kalman
f i l t e r a l g o r i t h m is a l s o e m p l o y e d f o r state
e s t i m a t i o n in MAMS. T h e p r e s e n c e of noise
affects t h e c o m m a n d e d a c c e l e r a t i o n through
t h e dish r a t e in P N , w h i l e i t a f f e c t s t h e m e a s u r e m e n t c o v a r i a n c e in APN a n d L Q g u i d a n c e
s c h e m e . T h e e s t i m a t i o n of t g o is e s s e n t i a l
in L Q g u i d a n c e s c h e m e . PN a n d APN d o n o t
need this estimate.
BORE SIGHT ERRCR
ANGULAR NOISE
- MISSILE BODY
AWE
GEOMEIW OF SEEKER
5. LQ Guidance
Maneuver detection and estimation
S t a r t i n g f r o m t h e l a t e 1960's, s e v e r a l
n e w e r f o r m s of g u i d a n c e laws using o p t i m a l
control and differential g a m e theories have
Modern guidb e e n proposed in l i t e r a t u r e [SI.
a n c e a n d e s t i m a t i o n t h e o r i e s would allow t h e
design t o "optimally" s e p a r a t e t h e signal f r o m
t h e n o i s e by using i n f o r m a t i o n a b o u t t h e missile
d y n a m i c s a n d noise c o v a r i a n c e s r a t h e r t h a n
f i l t e r i n g b a s e d only on f r e q u e n c y c o n t e n t s .
T h e e s t i m a t e s a r e t h e n used t o g e n e r a t e
t h e g u i d a n c e signal (i.e., c o m m a n d e d a c c e l e r a tion) b a s e d upon a s u i t a b l y s p e c i f i e d o p t i m a l i t y
c r i t e r i o n . This p e r f o r m a n c e index c a n b e s t r u c t u r e d t o i n c l u d e p e n a l t i e s o n u n d e s i r a b l e features
l i k e trajectory deviations, excessive
a c c e l e r a t i o n s etc. in a d d i t i o n t o miss d i s t a n c e .
A s i m p l i f i e d K a l m a n f i l t e r d e n o t e d as
SKF, w h i c h d o e s n o t c o n t a i n t h e m a n e u v e r
t e r m is basically u s e d f o r m a n e u v e r d e t e c t i o n
a n d e s t i m a t i o n . T h e m e t h o d used follows t h e
input e s t i m a t i o n p r o c e d u r e of C h a n e t a1 [9].
In t h i s s c h e m e , t h e d i f f e r e n c e b e t w e e n t h e
actual measurements and t h e s t a t e e s t i m a t e
f r o m t h e S K F is o b t a i n e d a t s u c c e s i v e m e a s u r e ment points t o e s t i m a t e the target maneuver.
Typically, t e n s a m p l e s of m e a s u r e m e n t s a r e
u s e d t o a r r i v e a t a n e s t i m a t e of t a r g e t maneuver. A w e i g h t e d l e a s t s q u a r e s moving window
a l g o r i t h m is a d o p t e d .
S u c c e s i v e v a l u e s of
t h e e s t i m a t e a r e used t o make a n assesment
of t h e o c c u r e n c e of t h e m a n e u v e r by c o m p a r ing t h e m with a p r e s e t t h r e s h o l d value. A
l a r g e d e p a r t u r e of m a n e u v e r e s t i m a t e f r o m
t h e t h r e s h o l d signals d e t e c t i o n . To e s t i m a t e
t h e l e v e l of t h e m a n e u v e r i t s e l f , a v a r i a n t
of t h e m e t h o d proposed in [91 is used. T h e
proposed m e t h o d u s e s i n c r e m e n t a l m e a s u r e m e n t s by c o n s i d e r i n g i n c r e m e n t s in t h e m e a s u r e m e n t s o v e r t h a t a t any c h o s e n t i m e point,
a f t e r t h e detection has occured.
The most commonly advocated approach
t o t a c t i c a l missile g u i d a n c e is t h e o n e b a s e d
on Linear-Quadratic-Gaussian
theory.
This
f o r m u l a t i o n r e q u i r e s t h e d e v e l o p m e n t of a
linear model for t h e engagement scenario
a n d s e l e c t i o n of a s u i t a b l e p e r f o r m a n c e index
t h a t r e s u l t s in a n e x p l i c i t f o r m f o r t h e solution.
Incorporating model-tracking features and
time-varying weights turns out t o be critical
f o r o b t a i n i n g miss d i s t a n c e a n d a c c e l e r a t i o n
p e r f o r m a n c e s u p e r i o r t o t h a t of PN a n d APN.
Engagement Conditions
Deterministic
212
and
stochastic
cases
are
b o t h c o n s i d e r e d . Only g l i n t noise i s included
in t h e s t o c h a s t i c case. P e r f e c t d e t e c t i o n a n d
e s t i m a t i o n implying n o d e t e c t i o n a n d e s t i m a t i o n
e r r o r s , a n d n o i s e - f r e e dish r a t e s a r e a s s u m e d
in t h e d e t e r m i n i s t i c runs. In t h e s t o c h a s t i c cases,
detection and estimation procedures described
e a r l i e r a r e used. Besides, t h e g l i n t n o i s e effect
o n t h e dish r a t e is a l s o considered. T h e c o m m a n d e d a c c e l e r a t i o n is o b t a i n e d by passing t h e dish
r a t e t h r o u g h a n o i s e f i l t e r in t h e s t o c h a s t i c
cases. Both t h e cases of s h o r t a n d long r a n g e
e n g a g e m e n t s a r e c o n s i d e r e d , with t h e t a r g e t
m a n e u v e r i n g at d i f f e r e n t points, t o bring o u t
t h e i r i n f l u e n c e on t h e a c c e l e r a t i o n p r o f i l e a n d
m i s s d i s t a n c e . A m a x i m u m v a l u e of 6" i s a r b i t r a rily s p e c i f i e d f o r t h e h e a d i n g e r r o r (HE). T h e
missile a c c e l e r a t i o n c a p a c i t y is a s s u m e d to
b e bounded. A f i x e d v a l u e of 8g i s u s e d f o r
t a r g e t accleration.
'1
-.
I
/ APN
MAMS
7. Comparative Performance
The main features t h a t a r e used for compa r i s o n in t h e d e t e r m i n i s t i c a n d s t o c h a s t i c cases
a r e ( a ) a c c e l e r a t i o n p r o f i l e s a n d (b) m i s s d i s t a n c e s
in P N , APN a n d L Q g u i d a n c e i m p l e m e n t a t i o n s .
c
10
0.5
0.25
0.75
1.0
time
FIG. 5
Accelerotion r e c p i r e m d comparison fcr early maneuver
Deterministic situation
( a ) T h e a c c e l e r a t i o n b e h a v i o u r in t h e d e t e r m i n i s t i c cases is s h o w n in F i g u r e s 5 a n d 6 f o r
early and l a t e maneuver occurence situations,
f o r z e r o heading e r r o r a n d long i n t e r c e p t range.It
is o b s e r v e d f r o m t h e f i g u r e s t h a t t h e a c c e l e r a t i o n is z e r o in all t h e cases till t h e o c c u r e n c e
of m a n e u v e r , s i n c e t h e heading e r r o r is z e r o .
When t h e m a n e u v e r o c c u r s , t h e c o m m a n d e d
a c c e l e r a t i o n in APN a n d MAMS show a s u d d e n
d i s c o n t i n u i t y unlike in PN. This i s b e c a u s e t h e
t a r g e t m a n e u v e r itself c h a n g e s a b r u p t l y f r o m
0 to 8g a n d in t h e APN a n e x t r a t e r m i s a d d e d
in ac t o t a k e a c c o u n t of t h e t a r g e t m a n e u v e r ,
w h i l e in MAMS ac is o b t a i n e d by using a pulsed
a c c e l e r a t i o n f o r t r a c k i n g , which is i n t r o d u c e d
suddenly b e c a u s e of t a r g e t maneuver.
MKS DISTANCES (met)
CONDITIONS
PN
APN
MAMS
A
=L.o
= 1.1
= 0.3
0
This way, in APN a n d MAMS a l a r g e a m o u n t
of a c c e l e r a t i o n is a d d e d a t t h e s t a r t of t a r g e t
m a n e u v e r itself. It is a d v a n t a g e o u s t o t u r n t h e
missile when t h e s e p a r a t i o n is l a r g e , s i n c e t h i s
would n e e d s m a l l p i t c h i n g motions. MAMS t a k e s
a d v a n t a g e of t h e p r e d i c t e d point of i n t e r c e p t
a n d s e t t l e s t o a s t r a i g h t l i n e c o u r s e . APN also
h a s a n e a r similar behaviour e v e n t h o u g h t h e r e
IS n o c o n c e p t
of a p r e d i c t e d i n t e r c e p t p o i n t
in APN. T h e c o m m a n d e d a c c e l e r a t i o n in P N
shows n o s u c h d i s c o n t i n u i t y s i n c e t h e t a r g e t
m a n e u v e r d o e s n o t c a u s e d i s c o n t i n u o u s dish
r a t e changes. PN shows l a r g e v a l u e s of a c w i t h
c h a n g e of sign, s i n c e t h e a c c e l e r a t i o n follows
t h e dish m o t i o n . T h e v a r i a t i o n in a c c e l e r a t i o n
a f t e r t h e c o m p l e t i o n of t h e pulse in APN IS
m u c h s m a l l e r , w h i l e t h a t in MAMS is t h e s m a l l est. I t is i n t e r e s t i n g t o o b s e r v e t h a t t h e APN
a c c e l e r a t i o n p r o f i l e r e s e m b l e s t h e MAMS p r o f i l e
closely.
The small acceleration
requirement
in APN m a k e s t h e blind p h a s e of t h e flight
e a s y t o c o p e , a s in MAMS.
FIG. 6
a2
0.4
0.6
0.8
lime
Accdemtion requimment comparison
for late manewer
In APN, t h e c o n t r i b u t i o n initially of t h e
t a r g e t acceleration t e r m t o t h e commanded
a c c e l e r a t i o n i s of t h e s a m e sign as t h a t of dish
r a t e ( b o t h being negative). H e n c e ac is a l a r g e
n e g a t i v e q u a n t i t y in APN, r e a c h i n g s a t u r a t i o n
value. In PN, t h e a c c e l e r a t i o n level u n d e r g o e s
l a r g e excursions. F u r t h e r , t h e APN a c c e l e r a t i o n
is m o r e u n i f o r m a r o u n d z e r o d u e t o t h e c o m b i n a t i o n of dish r a t e a n d t h e c o m p o n e n t of t a r g e t
a c c e l e r a t i o n , with t h e a c c e l e r a t i o n p r o f i l e of
MAMS r e p r e s e n t i n g t h e ideal s i t u a t i o n . Thus
t h e i n t e g r a t e d s q u a r e a c c e l e r a t i o n is s m a l l e r
in A P N c o m p a r e d t o PN.
T h e p r e s e n c e of i n i t i a l h e a d i n g e r r o r m o d i f i e s
t h e i n i t i a l p o r t i o n s of t h e a c c e l e r a t i o n profiles.
213
In P N and APN i t a d d s a positive a c c e l e r a t i o n
initially, while in MAMS a positive initial pulsed
a c c e l e r a t i o n is c a u s e d by HE. HE h o w e v e r h a s
n o significant effect o n t h e miss d i s t a n c e , p a r t i c u l a r l y f o r l a t e m a n e u v e r initiations.
(b) The m e a n a n d I C miss d i s t a n c e v a l u e s o b t a i n e d f o r P N , APN, a n d MAMS a r e r e c o r d e d in
T a b l e s 1 a n d 2 f o r s h o r t a n d long r a n g e cases
r e s p e c t i v e l y , with a heading e r r o r of 6". Even
w h e n p e r f e c t i n f o r m a t i o n is a s s u m e d a b o u t t h e
t a r g e t maneuver, t h e situation becomes stochastic
d u e to t h e noisy dish r a t e s . F r o m b o t h t h e cases,
it may b e observed t h a t t h e m a x i m u m v a l u e s
of miss which o c c u r in P N f o r l a t e m a n e u v e r
cases a r e significantly r e d u c e d in APN. F o r e a r l y
m a n e u v e r s , t h e APN miss d i s t a n c e s t e n d t o b e
l a r g e r t h a n in P N in t h e s h o r t r a n g e case. In
t h e long r a n g e case, v e r y e a r l y i n i t i a t i o n of
m a n e u v e r c a u s e s a l a r g e r miss t h a n in PN. This
is because, t h e a b s e n c e of t h e t a r g e t a c c e l e r a t i o n
t e r m in APN c a u s e s a n a b r u p t c h a n g e in ac w h e n
t h e t a r g e t l e v e l s off f o r r e t r e a t . C o m p a r i s o n
of t h e miss d i s t a n c e s in PN a n d APN f o r t h e
d e t e r m i n i s t i c a n d s t o c h a s t i c cases r e v e a l s t h a t
t h e y i n c r e a s e significantly - nearly by a f a c t o r
of 4 - in t h e s t o c h a s t i c cases. C o m p a r i s o n is
m a d e with MAMS only f o r t h e c r i t i c a l P N cases.
T h e miss d i s t a n c e values shown f o r MAMS a r e
only f o r t h e a s s u m e d m e a s u r e m e n t c o v a r i a n c e s .
For t h e o t h e r cases, t h e miss d i s t a n c e s in MAMS
a r e s m a l l e r t h a n t h e PN a n d APN values.
(b) T h e miss d i s t a n c e s f o r t h e s e e n g a g e m e n t
c o n d i t i o n s a r e a l s o shown in Figs 5 a n d 6. A
s t u d y of t h e miss d i s t a n c e s for various tman
r e v e a l s t h a t in PN, t h e miss d i s t a n c e is l a r g e
f o r both s m a l l and l a r g e v a l u e s of t
being v e r y s m a l l in b e t w e e n t h e s e v a ues. while
Also,
f o r excessively delayed maneuvers a n d for level
f l i g h t conditions, t h e miss d i s t a n c e s a r e very
small. APN a l s o shows a similar p e r f o r m a n c e
b u t w i t h m a x i m u m v a l u e s of m i s s s m a l l e r t h a n
t h o s e in PN. T h e miss d i s t a n c e s in MAMS could
b e r e d u c e d t o s m a l l v a l u e s in a l l t h e cases, by
p r o p e r l y tuning t h e p e r f o r m a n c e index w e i g h t s
a n d s e l e c t i o n of r e f e r e n c e t r a j e c t o r y .
Tan'
Stochastic situation
While considering t h e s t o c h a s t i c c a s e s , t h e
r e s u l t s h a v e b e e n a v e r a g e d o v e r 50 m o n t e c a r l o
runs. The point of m a n e u v e r d e t e c t i o n a n d t h e
m a n e u v e r level vary f r o m r u n t o run. Mean a n d
l a v a l u e s a r e shown in T a b l e s I a n d 2. I t is s e e n
f r o m t h e individual r u n s t h a t t h e d e t e c i o n delay
v a r i e s b e t w e e n 240 millisec and 500 millisec,
t h e delay being l a r g e r f o r s m a l l m a n e u v e r levels.
The a c c u r a c y of m a n e u v e r e s t i m a t i o n is o b s e r v e d
t o b e b e t t e r t h a n 6% in a l l t h e s e runs, f o r t h e
a s s u m e d m e a s u r e m e n t noise covariance.
8. Conclusions
As f a r as t h e miss d i s t a n c e is c o n c e r n e d ,
t h e m a x i m u m v a l u e o b t a i n e d in APN is s m a l l e r
t h a n in P N f o r t h e r a n g e of m a n e u v e r i n i t i a t i o n
t i m e s investigated. T h e miss d i s t a n c e s in t h e
L Q guidance s c h e m e c a n be made even smaller
by proper s e l e c t i o n of p e r f o r m a n c e index weig h t s a n d t h e r e f e r e n c e nominal.
(a) T h e a c c e l e r a t i o n profiles in PN, APN a n d
MAMS h a v e t h e s a m e p a t t e r n as in t h e d e t e r m i n i s t i c cases, b u t now show e x c u r s i o n s a r o u n d
t h e d e t e r m i n i s t i c values d u e t o t h e g l i n t noise
a f f e c t i n g c o m m a n d e d a c c e l e r a t i o n . The m a g n i t u d e
of t h e s e f l u c t u a t i o n s a r o u n d t h e d e t e r m i n i s t i c
profile increases terminally, d u e t o t h e glint
noise b e c o m i n g m o r e p r e d o m i n a n t t o w a r d s t h e
e n d of t h e e n g a g e m e n t .
In t h e d e t e r m i n i s t i c s t u d y , with t h e t a r g e t
m a n e u v e r i n i t i a t i o n o c c u r i n g a t various points
in t i m e , t h e P N a c c e l e r a t i o n p r o f i l e s exhibit
l a r g e m a g n i t u d e e x c u r s i o n s of b o t h p o l a r i t i e s
w h e r e a s in APN, t h e a c c e l e r a t i o n p r o f i l e r e s e m b l e s a pulse, with t h e pulse height being t h e
m a x i m u m permissible a c c e l e r a t i o n . F u r t h e r m o r e ,
TABLE 1. MISS DISTANCE COMPARISON (SHORT RANGE CASE)
HE
=
bo
N' = 5
HISS DISTANCE VAIUES (met)
L Q Guidance Scheme
APN
tman
PN
Perfect target
information
0.0
2.7 ( 2 . 2 )
3.90 ( 2 . 8 0 )
w i t h t a r g e t maneuver d e t e c t i o n anc
estimatior
(MAMS w i t h d e t e c t i on a n d e s t i m a t i o n )
3.60 ( 2 . 7 )
1 .o
2.4 ( 2 . 3 )
3.60 ( 2 . 9 6 )
3.70 ( 3 . 4 )
2.0
2.6 ( 2 . 5 )
5.40 ( 3 . 1 0 )
3.10 ( 2 . 6 )
2.5
10.9 ( 3 . 2 )
3.60 ( 3 . 1 4 )
5.40 ( 3 . 4 )
3.0
13.7 ( 4 . 3 )
4.00 ( 2 . 8 6 )
4.90 ( 3 . 0 )
level
4.3 ( 1 . 7 )
3.17 ( 2 . 5 0 )
3.17 ( 2 . 5 )
214
4.16 ( 2 . 0 7 )
TABIE 2 . MISS DISTAVCE COMPARISON (LONG EANGE CASE)
HE
= 6'
N' = 5
MISS DISTANCE VALUES (met)
I Q Guidance Scheme
APN
tinan
PN
Perfect t a r g e t
information
-
with t a r g e t maneuver d e t e c t i o n and
estimation
~~
0.0
4.8 ( 3 . 8 )
8.67 ( 4 . 8 0 )
12.85 ( 7 . 3 )
1 .o
9.9 ( 4 . 0 )
4.40 ( 3 . 1 8 )
5.43 ( 4 . 0 )
2.0
6.8 ( 3 . 7 )
3.60 ( 2 . 9 0 )
4.28 ( 2 . 7 )
5.0
2.8 ( 2 . 0 )
3.50 ( 2 . 6 0 )
3.10 ( 2 . 2 )
7.0
16.1 ( 5 . 0 )
3.70 ( 2 . 7 5 )
7.30 ( 3 . 7 )
7.5
18.3 ( 5 . 0 )
6.83 ( 2 . 9 4 )
4.62 ( 3 . 0 )
level
4.3 ( 2 . 8 )
3.80 ( 2 . 5 0 )
3.80 ( 2 . 5 )
2.13 ( 2 . 6 8 )
G u i d a n c e and C o n t r o l
Aug 1984, p p 78-98.
t h e i n t e g r a t e d a c c e l e r a t i o n s q u a r e is m u c h smaller in APN. This APN a c c e l e r a t i o n profile very
closely r e s e m b l e s t h e o n e o b t a i n e d f r o m t h e
L Q t r a c k i n g f o r m u l a t i o n of g u i d a n c e problem.
While t h e heading e r r o r modifies t h e initial accel e r a t i o n profile, i t h a s n o pronounced effect
o n t h e miss distance.
conference
proceedings,
4. Rogers, R.M., 'Sensitivity of higher o r d e r
g u i d a n c e law to p a r a m e t e r variations',
IEEE NAECON Proceedings, 1982.
5. F i t z g e r a l d
T.J., 'Shaping F i l t e r s f o r Dist r u b a n c e s with R a n d o m s t a r t i n g times',
AIAA J. G u i d a n c e a n d C o n t r o l , Vol. 2,
NO. 2, 1979, pp 152-154.
The r e s u l t s establish t h a t t h e a c c e l e r a t i o n
p r o f i l e s a r e q u a l i t a t i v e l y d i f f e r e n t under t h e
t w o dimensional e n g a g e m e n t s c e n a r i o c o m p a r e d
t o t h e o n e dimensional case p r e s e n t e d by Nesl i n e a n d Zarchan. While APN a n d L Q guidance
s c h e m e s involve additional e f f o r t in e s t i m a t i n g
t h e t a r g e t a c c e l e r a t i o n reasonably a c c u r a t e l y ,
t h e s m a l l e r i n t e g r a t e d a c c e l e r a t i o n c a n b e exploite d f o r r a n g e advantage.
6. S a r m a , I.G.
a n d Swamy, K.N.,
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215